File:Complex log mapping.svg
Summary
| Description |
English: Complex log mapping maps radii to horizontal lines and circles to vertical lines.
The principal branch , viewed as a mapping from to the horizontal strip defined by , has the following properties, which are direct consequences of the formula in terms of polar form:
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| Date | |
| Source | Own work |
| Author | Soul windsurfer |
| Other versions |
|
| Source code | Created using Maxima.
/*
b batch file for maxima
There are 2 complex planes :
* w-plane
* z-plane
z is a pont of z-planes
zz is a list of z points
zzz is a list of zz lists
1. draw cufves on z plane
2. draw images of z-curves under h on w plane
3. draw images of w-curves under hi on z plane
http://math.stackexchange.com/questions/114733/mapping-half-plane-to-unit-disk
Adam Majewski
*/
kill(all);
remvalue(all);
display2d:false$
/* definitions of functions */
/* map from z to w : w=h(z) */
h(z):=if z=0.0 then -1000 else rectform(float(log(z))); /* map from z plane to w plane : w = h(z) */
/* converts complex number into list for draw package */
draw_format(z):= [float(realpart(z)),float(imagpart(z))];
GiveCirclePoint(t) := rectform(float(R*%e^(%i*t*2*%pi))); /* gives point of unit circle for angle t in turns */
GiveZRayPoint(R) := rectform(float(R*%e^(%i*tRay*2*%pi))); /* gives point of external ray for radius R and angle tRay in turns */
compile(all);
R_max: 5;
R_min: 0;
dR: R_max - R_min;
NumberOfRays: 10;
iMax: 100; /* number of points to draw */
print (" ============== compute =============== ")$
/* curve is a list of points joned by lines */
/* f_0 plane = z-plane */
circle_angles: makelist(i/(10*iMax), i, 0, 10*iMax-1)$ /* more angles = more points */
/* Unit circle */
R: 1;
UnitCirclePointsZ: map(GiveCirclePoint, circle_angles)$
/* circle with radius 1/2 */
R: 0.5;
HalfUnitCirclePointsZ: map(GiveCirclePoint, circle_angles)$
/* External circles */
circle_radii: makelist(R_min+i, i, 1, dR);
ZCirclesPoints: [];
for R in circle_radii do
ZCirclesPoints: append(ZCirclesPoints, map(GiveCirclePoint, circle_angles))$
/* External W rays */
ray_radii: makelist(R_min+dR*i/iMax, i, 0, iMax)$
ray_angles: makelist(i/NumberOfRays, i, 0, NumberOfRays-1)$
ZRaysPoints: [];
for tRay in ray_angles do
ZRaysPoints: append(ZRaysPoints, map(GiveZRayPoint, ray_radii))$
/* --------- map from z to w plane using h -------------------------------*/
/* w-curves = images of z-circles */
WCirclesPoints: map (h, ZCirclesPoints)$
WRaysPoints: map (h, ZRaysPoints )$
UnitCirclePointsW : map(h,UnitCirclePointsZ)$
HalfUnitCirclePointsW : map (h, HalfUnitCirclePointsZ)$
/* w-curves = images of z-rays */
print ("-------------- convert lists of complex points to draw format lists ----------- ")$
UnitCirclePointsZ : map (draw_format, UnitCirclePointsZ)$
HalfUnitCirclePointsZ : map ( draw_format, HalfUnitCirclePointsZ)$
ZCirclesPoints:map(draw_format,ZCirclesPoints)$
ZRaysPoints:map(draw_format,ZRaysPoints)$
WCirclesPoints:map(draw_format, WCirclesPoints)$
WRaysPoints:map(draw_format, WRaysPoints)$
UnitCirclePointsW : map (draw_format, UnitCirclePointsW)$
HalfUnitCirclePointsW : map(draw_format, HalfUnitCirclePointsW)$
/* images of z-lines */
print (" ------------------ draw ------------------------------------------------------ ")$
path:"~/Dokumenty/maxima/log_map/"$ /* pwd */
FileName:"i"$ /* without extension which is the terminal name */
load(draw); /* Mario Rodríguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html */
draw(
terminal = 'svg,
file_name = concat(path,FileName),
columns = 2,
dimensions=[1000,500], /* x = y*columns */
gr2d(title = " z plane ",
/*
yrange = [-3,3],
xrange = [-3,3],
*/
points_joined =true,
grid = false,
point_size = 0.2,
point_type = filled_circle,
color = green,
points(ZCirclesPoints),
color = red,
points(UnitCirclePointsZ),
color = yellow,
points(HalfUnitCirclePointsZ),
color = blue,
points(ZRaysPoints)
),
gr2d(
title = " w plane: w=log(z)",
yrange = [-2.0,2.0],
xrange = [-2.0,2.0],
grid = false,
xaxis = false,
points_joined =true,
color = red,
point_size = 0.2,
point_type = filled_circle,
color = green,
points(WCirclesPoints),
color = red,
points(UnitCirclePointsW),
color = yellow,
points(HalfUnitCirclePointsW),
color = blue,
points(WRaysPoints)
)
);
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![[1]](./File:Complex_log_mapping.svg#cite_note-1){kind=link}
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
- ↑ Strictly speaking, the point on each circle on the negative real axis should be discarded, or the principal value should be used there.
